1438 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



Table XII — Derivation of the Minimum Sums 

 FOR the Transmission 



T = E (0, 3, 4, 5, 6, 7, 8, 10, 11) 



(a) Table of Prime Implicants 



Xi Xi 

 XiX%X\ 



X3X2X1 



Xi'XiXi 

 Xz'x^Xx 



XiXa'Xi 



A 

 B 

 C 

 D 

 E 

 F 



(b) Boolean Representation of Table 

 (B + C)(D + E)(A + B)(A)(A)(A + D)(C)(F)(E + F) 



(c) Consistent Row Sets 



(A, C, F, D), (A, C, F, E) 



T = Xi'xz + xz'xi'xi' + xaz'x2 + a;4'x2a;i 



T = Xa'X3 + Xs'Xi'Xi' + X^Xz'Xi + Xi'XiXx 



"or" (non-exclusive) and multiplication signifies 



addition stands for 

 "and." This expression can be interpreted as a Boolean Algebra expres- 

 sion and the Boolean Algebra theorems used to simplify it. In particular 

 it can be "multiplied out": 



(B + C) (D + E) = BD + BE + CD + CE 



This form is equivalent to the statement that columns and 3 require 

 that any consistent row set must contain either rows B and D, or rows 

 B and E, or rows C and D, or rows C and E. 



The complete requirements for a consistent row set can be obtained 

 directly by providing a factor for each column of the table. Thus for 

 Table XII the requirements for a consistent row set can be written as: 



(B + C)(D + E)(A + B)(A)(A)(A + D)(C)(F)(E + F) 



By using the theorems that A-(A + D) = A and A- A = A, this can 

 be simplified to ACF(D + E). Thus the two consistent row sets for this 

 table are A, C, F, D and A, C, F, E and since they both contain the 

 same number of rows, they both represent minimum sums. This is true 

 only because rows D and E contain the same number of crosses. In 

 general, each row should be assigned a weight w = n — \og,2k, where 

 n is the number of variables in the transmission being considered and 



